Weak Hopf Algebras I: Integral Theory and C^*-structure

We give an introduction to the theory of weak Hopf algebras proposed recently as a coassociative alternative of weak quasi-Hopf algebras. We follow an axiomatic approach keeping as close as possible to the "classical" theory of Hopf algebras. The emphasis is put on the new structure related to the presence of canonical subalgebras A^L and A^R in any weak Hopf algebra A that play the role of non-commutative numbers in many respects. A theory of integrals is developed in which we show how the algebraic properties of A, such as the Frobenius property, or semisimplicity, or innerness of the square of the antipode, are related to the existence of non-degenerate, normalized, or Haar integrals. In case of C^*-weak Hopf algebras we prove the existence of a unique Haar measure h in A and of a canonical grouplike element g in A implementing the square of the antipode and factorizing into left and right algebra elements. Further discussion of the C^*-case will be presented in Part II.Comment: 40 pages, LaTeX, to appear in J. Algebr

Quantum Einstein's Equations and Constraints Algebra

In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein equations. We shall investigate this problem in the de-Broglie--Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein's equations are derived.Comment: 13 pages, No figure, RevTeX. To appear in Pramana J. Phys., 200

The Higgs Boson Mass in Split Supersymmetry at Two-Loops

The mass of the Higgs boson in the Split Supersymmetric Standard Model is calculated, including all one-loop threshold effects and the renormalization group evolution of the Higgs quartic coupling through two-loops. The two-loop corrections are very small (<<1 GeV), while the one-loop threshold corrections generally push the Higgs mass down several GeV.Comment: 17 pages. 4 figures. Improved discussion and notation. Corrected typos. Added references. Added plots. Main results unchange

Relativistic Partial Wave Analysis Using the Velocity Basis of the Poincare Group

The velocity basis of the Poincare group is used in the direct product space of two irreducible unitary representations of the Poincare group. The velocity basis with total angular momentum j will be used for the definition of relativistic Gamow vectors.Comment: 14 pages; revte

Relativistic tunneling through opaque barriers

We propose an analytical study of relativistic tunneling through opaque barriers. We obtain a closed formula for the phase time. This formula is in excellent agreement with the numerical simulations and corrects the standard formula obtained by the stationary phase method. An important result is found when the upper limit of the incoming energy distribution coincides with the upper limit of the tunneling zone. In this case, the phase time is proportional to the barrier width.Comment: 11 pages, 3 figure

Exact Dynamical and Partial Symmetries

We discuss a hierarchy of broken symmetries with special emphasis on partial dynamical symmetries (PDS). The latter correspond to a situation in which a non-invariant Hamiltonian accommodates a subset of solvable eigenstates with good symmetry, while other eigenstates are mixed. We present an algorithm for constructing Hamiltonians with this property and demonstrate the relevance of the PDS notion to nuclear spectroscopy, to quantum phase transitions and to mixed systems with coexisting regularity and chaos.Comment: 10 pages, 5 figures, Proc. GROUP28: The XXVIII Int. Colloquium on Group-Theoretical Methods in Physics, July 26-30, 2010, Newcastle upon Tyne, U

On the Perturbative Solutions of Bohmian Quantum Gravity

In this paper we have solved the Bohmian equations of quantum gravity, perturbatively. Solutions up to second order are derived explicitly, but in principle the method can be used in any order. Some consequences of the solution are disscused.Comment: 14 Pages, RevTeX. To appear in Phys. Rev.

Classical mechanics without determinism

Classical statistical particle mechanics in the configuration space can be represented by a nonlinear Schrodinger equation. Even without assuming the existence of deterministic particle trajectories, the resulting quantum-like statistical interpretation is sufficient to predict all measurable results of classical mechanics. In the classical case, the wave function that satisfies a linear equation is positive, which is the main source of the fundamental difference between classical and quantum mechanics.Comment: 11 pages, revised, to appear in Found. Phys. Let

Winterberg's conjectured breaking of the superluminal quantum correlations over large distances

We elaborate further on a hypothesis by Winterberg that turbulent fluctuations of the zero point field may lead to a breakdown of the superluminal quantum correlations over very large distances. A phenomenological model that was proposed by Winterberg to estimate the transition scale of the conjectured breakdown, does not lead to a distance that is large enough to be agreeable with recent experiments. We consider, but rule out, the possibility of a steeper slope in the energy spectrum of the turbulent fluctuations, due to compressibility, as a possible mechanism that may lead to an increased lower-bound for the transition scale. Instead, we argue that Winterberg overestimated the intensity of the ZPF turbulent fluctuations. We calculate a very generous corrected lower bound for the transition distance which is consistent with current experiments.Comment: 7 pages, submitted to Int. J. Theor. Phy

Gauge-Invariant Formulation of Spin-Current-Density Functional Theory

Spin-currents and non-abelian gauge potentials in electronic systems can be treated by spin-current-density functional theory, whose main input is the exchange-correlation (xc) energy expressed as a functional of spin-currents. Constructing a functional of spin currents that is invariant under U(1)$\times$SU(2) transformations is a long-standing challenge. We solve the problem by expressing the energy as a functional of a new variable we call "invariant vorticity". As an illustration we construct the xc energy functional for a two-dimensional electron gas with linear spin-orbit coupling and show that it is proportional to the fourth power of the spin current.Comment: 8 pages, 3 figures, submitte